Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual space to the smooth densities. Let $A$ be a pseudodifferential operator $A:C_0^{\infty}(M)\to C^{\infty}(M)$.
We start by noting that we may imbed $C^{\infty}$ in $\mathscr{D}'(M)$ by $\int_M uv$, where $u\in C^{\infty}$ and $v$ is a density with compact support, and thereby also $C_0^{\infty}(M)$ in $\mathcal{E}'(M)$. We also note that $C_0^{\infty}(M)$ is dense in $\mathcal{E}'(M)$. We wish to extend $A$ by continuity to an operator $A: \mathcal{E}'(M)\to \mathscr{D}'(M)$.
What I've been thinking is that if $(u_n)_{n\in \mathbb{N}}$ is a sequence converging to some $u\in \mathcal{E}'(M)$, then we wish to show that $Au_n$ converges to some $v\in \mathscr{D}'(M)$, i.e. $\langle Au_n,\phi \rangle$ converges to $\langle v,phi \rangle$ for every $\phi$ a smooth density with compact support, since $\mathscr{D}'(M)$ is equipped with the weak$^*$ topology. However, I seem to be stuck at this point, so any hint or idea at all would be nice.
EDIT:
I just realized that I (think) just need to show continuity on $C_0^{\infty}(M)$ since it is dense in $\mathcal{E}'(M)$. So if we let $(u_n)_{n\in \mathbb{N}} \to u \in C_0^{\infty}$ we have $\langle Au_n, \phi \rangle \to \langle Au ,\phi \rangle$ for all $\phi \in C_0^{\infty}(M)$, and therefore $A:\mathcal{E}'(M) \to \mathscr{D}'(M)$ is continuous on $C_0^{\infty}$, and thereby also on $\mathcal{E}'$ since $C_0^{\infty} is dense as mentioned.
